Protein Dimerization Network
Multi-input, Cell-type-specific Computation
Cell Type 1
Cell Type 2
Input 1
Input 2
Output
Input 1
Input 2
Output
Tune Expression
Levels
Output
Dimer
Input 1
Input 2
Input 1
Input 2
Output
Dimer
Rational or Random
A
C
B
Estrogen
Receptor α
Estrogen
Receptor β
DAX1
Glucocorticoid
Receptor
Androgen
Receptor
SHP
Many-to-Many
Dimerization
Dimerizing Transcription Factors
Example: Nuclear Hormone Receptors
Transcription
factor activity
Various output genes
Inputs:
Monomers
Outputs:
Dimers
Signal 2
Signal 1
Signal 3
Cell-Type-Specific
Expression
E
Size
3 monomers
12 monomers
7 edges
28 edges
Connectivity
F
Dependence on
Network Properties
D
Reactions
Monomers
Dimers
Diagram
Input-Output Function
=
Monomer Abundance
low
high
strong
weak
Dimerization Affinity ( )
+
+
+
+
Input M1
Output D22
Accessories:
M2, M3
M2
M3
Versatility:
All functions possible with fixed interactions
Expressivity:
Range of functions possible across all networks
Input-Output
Functions
Networks
Different
interactions
Same interactions,
different expression
Input: Total [
M1
]
Outputs:
Dimers
(e.g.,
D22
)
1
10
100
Leydig cells
ERα
AR
GR
ERβ
SHP
DAX1
1
10
100
Late spermatids
mRNA Expression
A
B
C
D
F
G
H
E
Two-input
Bump Function
M1 NOT M2
(NIMPLY)
NAND Gate
XOR Gate
Switch On
Input M1
Output
D33
Output
D22
Switch Off
Input M1
Bump Function
Input M1
Output
D33
Inverted Bump
Output
D55
Input
M2
Input
M1
M3
M4
Input M1
Output
D44
Output
D13
Input
M2
Input
M1
M3
M4
Input
M2
Input
M1
Output
D34
M3
M4
Input M1
Input M2
Input M1
Input M1
Input M1
Concentration
Input M1
10
-3
10
-1
10
-1
10
-3
10
3
1
10
1
Concentration
Input M1
10
-3
10
-3
10
3
1
1
10
-2
Input M1
10
-3
10
-3
10
3
1
1
10
-2
10
-1
10
-1
Input M1
10
-3
10
-3
10
3
1
10
1
Response
of D22
Response
of D33
Response
of D33
Response
of D44
10
-3
10
3
1
10
-2
10
-3
10
-1
10
-3
10
3
Response
of D66
10
-3
10
3
1
10
-2
10
-3
10
-1
1
10
-3
10
3
Response
of D55
10
-3
10
3
1
10
-1
10
-3
10
1
10
-3
10
3
Response
of D34
10
-3
10
3
1
10
-1
10
-3
10
1
1
1
1
1
10
-3
10
3
Response
of D13
M2
M2
M3
M4
M2
M3
M2
M3
M3
M5
M6
M4
Output
D66
Input
M2
Input
M1
M5
Input M1
Output D33
Input M1
Output D33
Input
M1
Input
M2
Output
D37
A
B
C
D
Network
Cell-Type Specific
Accessory Expression
Cell-Type Specific
Responses
Input
M2
Input
M1
Output D55
M2
M3
M2
M3
M3
M5
M4
M3
M4
M5
M6
M7
M8
Concentration
10
-1
10
1
10
3
M2
M3
Cell 1
M2
M3
Cell 2
M2
M3
Cell 3
Cell 1
Cell 2
Cell 3
10
-3
10
-2
10
-1
10
1
10
2
10
3
10
-1
1
1
Input M1
Concentration
Response of D33
Concentration
10
-3
10
1
10
-1
10
3
M2
M3
Cell 1
M2
M3
Cell 2
Cell 1
Cell 2
10
3
10
-3
1
10
-3
10
3
1
Concentration
Input M1
Response of D33
10
-1
10
1
10
-1
10
1
10
3
10
-1
10
1
10
3
10
3
Cell 1
Cell 2
Cell 3
Cell 4
Cell 5
Cell 6
M3
M4
M5
Cell 7
M3
M4
M5
Cell 8
M3
M4
M5
Cell 9
Concentration
Cell 1
M3
M4
M5
M6
M7
M8
Cell 2
Concentration
10
-3
10
1
10
-1
10
3
10
-3
10
1
10
-1
10
3
10
-1
10
1
10
-3
10
-2
10
-1
10
3
10
1
10
3
10
-1
1
2
3
4
5
6
7
8
9
Cell Types:
Input M1
Input M2
Response of D55
Concentration
10
-1
10
1
10
-3
10
-2
10
-1
10
3
10
1
10
3
10
-1
Input M1
Input M2
Response of D37
Concentration
Cell 2
Cell 1
A
B
Discretize Outputs &
Count Unique Functions
D
Generate & simulate
random networks
Network Size
Expressivity by
Network Size
“Function A”
“Function B”
3 monomers
12 monomers
Generate & simulate
random networks
Sparse
Dense
C
Minimum Network Sizes Required for Select Functions
Network Size
6
3
5
4
2D Input
Input M1
Output
Input M2
Input M1
Output
1D Input
Network Size
3
4
5
6
7
8
9
10
11
12
2
Unique Discretized Functions
1D Input
(6 x 6 grid)
2D Input
(6 x 6 x 6 grid)
10
3
10
-3
1
10
-3
10
3
1
Input M1
Output
0
200
400
600
2
4
6
8
10
12
0
2000
4000
6000
Network Connectivity
Network Connectivity
Unique Functions
Unique 1-input Functions
Unique 2-input Functions
0%
25%
50%
75%
100%
0
1000
2000
3000
4000
5000
6000
7000
0%
25%
50%
75%
100%
0
100
200
300
400
500
600
700
A
C
F
B
E
D
Functi on
Mo
nomers
Required
"At least 1" gate
6
"At least 2" gate
7
"At least 3" gate
10
4-i nput AND gate
8
Output Concentration
10
-1
10
-2
10
-3
Responses
1
0
0
0
0
0
0
0
Input Monomer
M1
M2
M3
M4
Input Combinations
0
1
1
1
0
1
1
0
0
1
0
1
0
1
0
0
0
0
1
1
0
0
1
0
0
0
0
1
0
0
0
0
Inputs
Responses
1
1
1
0
1
0
0
0
M1
M2
M3
M4
Input Monomer
1
1
1
1
1
1
1
0
1
1
0
1
1
1
0
0
1
0
1
1
1
0
1
0
1
0
0
1
1
0
0
0
Inputs
“Exactly Any 2 or None”
3-input Logic Gate
Output
D46
Input
M1
Input
M3
Input
M2
Output Concentration
10
-1
10
-2
10
-3
Responses
0
1
1
0
1
0
0
1
Input Monomer
Inputs
M1
M2
M3
Input Combinations
1
1
1
1
1
0
1
0
1
1
0
0
0
1
1
0
1
0
0
0
1
0
0
0
Output
D68
“At least 3”
4-input Logic Gate
Input
M3
Input
M2
Input
M1
Input
M4
3
4
5
6
7
8
9
10
Network Size
0
10
20
30
40
54
Functions Achieved
Expressivity of all
3-input logic gates
G
2 Inputs
3 Inputs
4 Inputs
vs. 11 combined AND & OR gates...
vs. 11 combined AND & OR gates...
A
B
C
B
B
C
A
C
C
B
A
A
A
B
D
A
C
D
B
B
A
C
C
D
0
3
6
9
12
Number of AND/OR Gates Required
0
5
10
Required
Network Size
Scaling of Network Size
vs. Boolean Complexity